Lattice dynamics and correlated atomic motion from the atomic pair distribution function

INTRODUCTION

The pair distribution function ~PDF! obtained from thepowder x-ray and neutron diffraction experiments has beenshown to be of great value in determining the local atomic structure of materials.1The PDF results from a Fourier trans-form of the powder diffraction spectrum Bragg peaks 1diffuse scattering! into real-space.2For well ordered crystals,apart from technical details, this is similar to fitting theBragg peaks 1 thermal diffuse scattering in the powder pat-tern in a manner first discussed by Warren.3A PDF spectrumconsists of a series of peaks, the positions of which give thedistances of atom pairs in real space. The ideal width of these peaks ~aside from problems of experimental resolution! isdue both to relative thermal atomic motion and to static dis-order. Thus an investigation of the effects of lattice vibra-tions on PDF peak widths is important for at least two rea-sons: first, to establish the degree to which information onphonons and the interatomic potential! can be obtained frompowder diffraction data, and, second, to account for correla-tion effects in order to properly extract information on staticdisorder in a disordered system such as an alloy.

In general, powder diffraction is not considered a favor-able approach for extracting information about phononssince, not only is energy information lost in the measure-ment, but also the diffuse scattering is isotropically averaged.The lattice vibrations are best described from the phonondispersion curves determined using inelastic neutron scatter-ing and high-energy-resolution inelastic x-ray scattering onsingle crystals.4,5Nevertheless, with the advent of high-energy synchrotron x-ray and pulsed-neutron sources andfast computers, it is possible to measure data with unprec-edented statistics and accuracy. The PDF approach has beenshown to yield limited information about lattice vibrations in powders, though the extent of which this information can beextracted remains controversial.7–10Measuring powders has the benefit that the experimentsare straightforward and do not require single crystals. It isthus of great interest to characterize the degree to whichlattice vibrations are reflected in the PDF using simple mod-els, such as the Debye model, in situations where detailedinteratomic potential information is not available. In this pa-per we explore these issues by comparing both measuredPDFs and those calculated from realistic potential modelswith PDFs obtained through a single-parameter Debyemodel. This comparison is carried out as a function of atomicpair separation, temperature and direction in the lattice. Wefind that a single parameter Debye model explains much ofthe observed lattice vibrational effects on PDF peak widths,including the temperature dependence, in crystals like Ni,Ce, and GaAs. However, small but non-negligible deviationsfrom the Debye model calculation are evident in crystalwhich needs a long-range interaction to explain anomalies inthe dispersion curves.

CORRELATED ATOMIC MOTION IN REAL SPACE

The existence of interatomic forces in crystals results inthe motion of atoms being correlated. This is usually treatedtheoretically by transforming the problem to normal coordi-nates, resulting in normal modes ~phonons! that are non-interacting, thus making the problem mathematically tract-ible. Projecting the phonons back into real-space coordinatesyields a picture of the dynamic correlations. This situationcan be understood intuitively in the following way. Figure 1shows a schematic diagram of atomic motion in three differ-ent interatomic force systems, each with its correspondingideal PDF spectrum. In a rigid-body system Fig. 1-a, the

interatomic force is extremely strong and all atoms move inphase. In this case, the peaks in the PDF are delta-functions.At the opposite extreme the atoms are non-interacting ~theEinstein model! and move independently as shown in Fig.1~b!. This type of atomic motion results in broad PDF peakswhose widths are given by the root mean-square displace-ment amplitude (A^u2&). In real materials, the interatomicforces depend on atomic pair distances, i. e., they are strongfor nearest-neighbor interactions and get weaker as theatomic pair distances increase. In fact, these interactions areoften quite well described with just nearest-neighbor or first-and second-nearest-neighbor coupling. The case of nearest-neighbor interactions is shown in Fig. 1~c!. In this ~Debye!model a single parameter corresponding to the spring con-stant of the nearest-neighbor interaction is used. Here, near-neighbor atoms tend to move in phase with each other, whilefar-neighbors move more independently. As a result, thenear-neighbor PDF peaks are sharper than those of far-neighbor pairs. This behavior was first analyzed by Kaplowand co-workers in a series of papers11–13for a number ofelemental metals.

EXPERIMENTS AND ANALYSIS

The experimental PDFs discussed here were measured us-ing pulsed neutrons and synchrotron x-ray radiation. Theneutron measurements were carried out at the NPD diffrac-tometer at the Manual Lujan, jr., Neutron Scattering Center~LANSCE! at Los Alamos and the x-ray experiments atbeam line A2 at CHESS ~Cornell!. Powder samples of Niand a polycrystalline Ce rod were loaded into a vanadiumcan for the neutron measurements, carried out at room temperature. Powdered GaAs was placed between thin foils ofkapton tapes for the x-ray measurements, measured at 10 Kusing 60 KeV (l50.206 Å) x rays. Due to the higher x-rayenergy at CHESS and relatively low absorption coefficient ofGaAs, symmetric transmission geometry was used.Both the neutron and x-ray data were corrected14,15forexperimental effects and normalized to obtain the total scat-tering function S(Q), using programs PDFgetN.

DISCUSSION

The mean-square relative displacements sij2and the cor-responding correlation parameter shown in Figs. 2, 4, 5, 8,and 9 present two interesting pieces of information about the atomic motions in a crystalline material. First of all, theyshow that nearest-neighbor atomic motion is significantlycorrelated. Second, the details of the motional correlations asa function of pair distance display structures which deviatefrom the predictions of the simple CD model. Here we canraise some interesting questions. How is this structure in themotional correlation of atom pairs related to the underlyinginteratomic potentials? Can one extract the potential param-eters using an inverse approach to model the PDF peakwidths with the potential parameters as input?Reichardt and Pintschovius8argued that the calculatedPDF peak widths as a function of pair distance are ratherinsensitive to the details of the lattice dynamics models usedto calculate sij2. They found that PDFs calculated using ei-ther very simple or complex models didn’t show significantdifferences. A similar conclusion has been reached by Graf etal.,10in contradiction to previous claims by Dimitrov et al.7Indeed, the magnitude of errors implicit in the measurementand data analysis appear to be comparable to the effects thatmust be measured to obtain quantitatively accurate potentialinformation using this approach.9The conclusions of Rei-chardt and Pintschovius and Graf et al. and Thorpe et al. arelargely borne out by the present work; e.g., the grossly over-simplified CD model, which neglects elastic anisotropy andparameterizes the dynamics with a single number u ,is rather successful at explaining the smooth rijdependence ofthe PDF peak widths.Thus, when the BvK force parameters are not available,we have shown that the correlated Debye ~CD! model is areasonable approximation to describe both the smoothrij-dependence and the temperature dependence of sij2insimple elements. Considering the poor correspondence be-tween the Debye phonon density of states and the BvK den-sity of states, the reasonable agreement between the BvKmodel calculations of sij2and that of the CD model is rathersurprising. This confirms that the PDF peak width is ratherinsensitive to the details of the phonon density of states andthe phonon dispersion curves, as suggested by Reichardt andPintschovius and by Graf et al. Any information about theinteratomic forces in the PDF peak widths is contained in thesmall deviations of the sij2from those of the CD model cal-culations. Therefore, extracting interatomic potential infor-mation from the PDF peak widths is unlikely. However,these deviations could possibly yield some average phononinformation. For example, recent calculations by Graf et al.10showed that one can obtain phonon moments within a fewpercent accuracy for most fcc and bcc crystals using thenearest-neighbor force parameters extracted from a theoreti-cal BvK PDF spectrum. This result indicates that the PDFspectrum contains some average phonon information, although it doesn’t provide detailed phonon dispersion infor-mation. The average phonon information, such as phononmoments from the PDF peak widths, will be a usefulcomplement to optical and acoustic techniques that yieldzone-center information in situations where single crystalmeasurements are not possible. This complementarity alsoextends to the extraction of Debye-Waller factors from pow-der diffraction measurements.Finally, a comparison of the CD model calculations of the PDF peak widths in GaAs with those of experimental PDFand Kirkwood model calculations shows additional limita-tions of the CD model. In the CD model calculation, thenear-neighbor PDF peaks below r<5>